Statement

The number of linear representations of over with rational character values and for which no proper nonzero subrepresentation has rational character values.

The number of equivalence classes in under rational conjugacy.

The number of conjugacy classes of cyclic subgroups in .

Caveats

The number of irreducible representations over rationals is not the same as the number of irreducible representations over the complex numbers that can be realized over the rationals. The latter number is either smaller or equal, and it is equal when the group is a rational group, which means that any two elements generating the same cyclic subgroup are conjugate.

Also, although the counts in (1) and (2) are equal, it is possible for a rational character to arise from an irreducible representation over the complex numbers that is not realized over the rationals. However, some multiple of that representation can be realized over the rationals. This explains the equality of counts in (1) and (2). The smallest multiple used is termed the Schur index.

To prove: The number of irreducible representations of over the field of rational numbers (note: these need not be absolutely irreducible representations) equals the number of equivalence classes in under rational conjugacy (which can also be described as the number of conjugacy classes of cyclic subgroups).

Proof: As in the statement of Fact (2), we denote by the set of conjugacy classes of and by the set of irreducible representations of over a splitting field of characteristic zero.

No.

Assertion/construction

Facts used

Given data used

Previous steps used

Explanation

1

Let be the cyclotomic extension where is a primitive root of unity for the exponent of . Then, is a splitting field for in characteristic zero and hence the irreducible representations of over are all realized in .

Fact (1)

is a finite group

Fact-direct

2

Under the action of the group , the set of orbits in is in bijection with the set of irreducible representations of over .

Fact (2)

Apply Fact (2) to the extension over . Note that since it is a cyclotomic extension, it is automatically Galois.

3

We get an action of the group on induced as follows: for any element of that sends to , the induced permutation on sends the conjugacy class of to the conjugacy class of .

4

Under the action described in Step (3) for the group on , the set of orbits in is in bijection with the set of equivalence classes under rational conjugacy in

5

For every element , the cycle type of the permutation induced by on is the same as the cycle type of the permutation induced on as described in Step (3)

Fact (4) (Brauer's permutation lemma)

6

For every element , the number of fixed points of the permutation induced by on is the same as the number of fixed points of the permutation induced on as described in Step (3)

Step (5)

direct

7

The number of orbits of under the action of equals the number of orbits of under the action of .

By Fact (3), the number of orbits for a finite group action of a set can be computed in terms of the number of fixed points of each group element. Since these numbers are the same for both group actions, the number of orbits must also be the same).

8

The number of irreducible representations over rationals equals the number of equivalence classes under rational conjugacy

By Step (2), the number of irreducible representations over rationals = the number of orbits in under . By Step (4), the number of equivalence classes under rational conjugacy = the number of orbits in under . By Step (7), both number or orbits counts are equal. Combining, we get the desired result.